Chapter 6

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) To the nearest day, how long will it take for half of the Iodine-125 to decay?

For the following exercises, use the definition of a logarithm to solve the equation. \(5 \log _{7} n=10\)

For the following exercises, use properties of logarithms to evaluate without using a calculator. \(6 \log _{8}(2)+\frac{\log _{8}(64)}{3 \log _{8}(4)}\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{9}(x)=\frac{1}{2}\)

For the following exercises, describe the end behavior of the graphs of the functions. \(f(x)=3(4)^{-x}+2\)

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). An account is opened with an initial deposit of \(\$ 6,500\) and earns \(3.6 \%\) interest compounded semi-annually. What will the account be worth in 20 years?

For the following exercises, refer to Table \(8 .\) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array} $$ Use the regression feature to find an exponential function that best fits the data in the table.

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) Write an exponential model representing the amount of Iodine-125 remaining in the tumor after \(t\) days. Then use the formula to find the amount of Iodine- 125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

For the following exercises, use the definition of a logarithm to solve the equation. \(-8 \log _{9} x=16\)

For the following exercises, use properties of logarithms to evaluate without using a calculator. \(2 \log _{9}(3)-4 \log _{9}(3)+\log _{9}\left(\frac{1}{729}\right)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{18}(x)=2\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) 4\) units upward

For the following exercises, refer to Table \(8 .\) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array} $$ Write the exponential function as an exponential equation with base \(e\).

A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

For the following exercises, use the definition of a logarithm to solve the equation. \(4+\log _{2}(9 k)=2\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{3}(22)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{6}(x)=-3\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) \quad 3\) units downward

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Solve the compound interest formula for the principal, \(P\).

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

For the following exercises, use the definition of a logarithm to solve the equation. \(2 \log (8 n+4)+6=10\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{8}(65)\)

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log (x)\) and \(g(x)=10^{x}\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) \quad 2\) units left

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth \(\$ 14,472.74\) after earning \(5.5 \%\) interest compounded monthly for 5 years. (Round to the nearest dollar.)

For the following exercises, refer to Table \(8 .\) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array} $$ Use the intersect feature to find the value of \(x\) for which \(f(x)=250\)

For the following exercises, use the definition of a logarithm to solve the equation. \(10-4 \ln (9-8 x)=6\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{6}(5.38)\)

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log (x)\) and \(g(x)=\log _{\frac{1}{2}}(x)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\ln (x)=2\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) \quad 5\) units right

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (10-3 x)=\ln (-4 x)\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{4}\left(\frac{15}{2}\right)\)

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log _{4}(x)\) and \(g(x)=\ln (x)\)

For the following exercises, use the definition of common and natural logarithms to simplify. \(\log \left(100^{8}\right)\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Reflect \(f(x)\) about the \(x\) -axis

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Use properties of rational exponents to solve the compound interest formula for the interest rate, \(r\).

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Use the LOGarithm option of the REGression feature to find a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{13}(5 n-2)=\log _{13}(8-5 n)\)

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{\frac{1}{2}}(4.7)\)

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=e^{x}\) and \(g(x)=\ln (x)\)

For the following exercises, use the definition of common and natural logarithms to simplify. \(10^{\log (32)}\)

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Reflect \(f(x)\) about the \(y\) -axis

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Use the logarithmic function to find the value of the function when \(x=10\)

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. To the nearest whole number, what was the initial population in the culture?

For the following exercises, use the one-to-one property of logarithms to solve. \(\log (x+3)-\log (x)=\log (74)\)

Use the product rule for logarithms to find all \(x\) values such that \(\log _{12}(2 x+6)+\log _{12}(x+2)=2 .\) Show the steps for solving.